基于Jacobi-Davidson算法的大规模模态分析并行计算研究

对Jacobi-Davidson（J-D）算法进行了改进和并行计算研究。通过添加谱变换、收缩和重启动等策略将J-D算法改造成了适应大规模模态分析的算法。利用改进后的算法和各种数值求解软件包，建立了一套基于PANDA框架的模态分析并行求解体系。基于该求解体系和并行机群，开展了某工程结构大规模模态分析并行可扩展性研究，测试规模从数十万自由度一直达到千万自由度，并行CPU核数达到128个；研究了改进后的J-D算法内层迭代步数、重启动向量个数等控制参数对外层迭代收敛速度的影响；获取了不同规模并行计算的加速比。研究结果表明，改进后的J-D算法完全适应千万自由度规模以上的模态分析，内存占用与规模之间呈线性增长趋势，在1 025万自由度规模模态分析仅占用39.4 GB内存；同时该算法具有优异的并行可扩展性，在128个CPU测试核内接近线性加速，并且测试规模越大，曲线越接近理想加速曲线，1 025万自由度规模在128核的并行效率达到88.1 %。

PARALLEL COMPUTING STUDY OF LARGE-SCALE MODAL ANALYSIS BASED ON THE JACOBI-DAVIDSON ALGORITHM

Some improvements and parallel computing studies were carried out with the Jacobi-Davidson(J-D) method. Some strategies, such as the spectral transformation technique, restart and deflation techniques, were added and integrated with the J-D method to make it suitable to the large-scale modal analysis. A parallel modal solving system based on PANDA framework was created using the improved J-D algorithm and various numerical software packages. Utilizing the solving system and parallel computers, the parallel scalability of the J-D algorithm was studied via numbers of tests on an engineering structure. The maximum computing scale is over 10 million degrees of freedom, and the maximum numbers of parallel CPU processors attain 128. The influences of inner iteration steps and numbers of restarted vectors on the convergence velocity of outer iterations were studied, and the speedup curves for different scales were obtained. The results show that the improved J-D method is competent for the large-scale modal analysis, the memory costs increase linearly with the computing scales and only 39.4 GB of memory is needed for the modal analysis of 10.25 million scales. Also, the improved J-D method takes on an excellent parallel scalability that the speedup curves are almost linear within 128 testing processors and the curve is gradually close to the ideal speedup one as the accretion of computing scales. The parallel efficiency of 10.25 million scales at 128 processors attain 88.1 %.